Community Inference From Partially Observed Graph Signals: Algorithms and Analysis

This paper considers community inference methods for finding communities on a graph. We treat the setting where the edges are not fully observed. Instead, inference is based on partially observed filtered graph signals where observations from some nodes are missing. Under this setup, we treat two related tasks:
$\mathsf{A}$
)
blind
inference which recovers the inherited communities on the sub-graph;
$\mathsf{B}$
)
semi-blind
inference which recovers communities on the full graph with additional partial topology information. For task
$\mathsf{A}$
, we suggest a spectral method which analyzes the principal components of the data covariance matrix. We prove that it succeeds in finding the ‘true’ communities if the graph filter is low-pass and the nodes are uniformly sampled. For task
$\mathsf{B}$
, we propose a method using spectral interpolation with a Nyström extension. The latter approach is proven to succeed in finding the ‘true’ communities for modular graphs and low-pass graph filters. Numerical experiments on synthetic and real data corroborate our results.